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arXiv · 2308.10100

Catalan numbers: from FC elements to classical diagram algebras

Sadek Al Harbat
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Abstract

Let $W^c(A_n)$ be the set of fully commutative elements in the $A_n$-type Coxeter group. Using only the settings of their canonical form, we recount $W^c(A_n)$ by the recurrence that is taken as a definition of the Catalan number $C_{n+1}$ and we find the Narayana numbers as well as the Catalan triangle via suitable set partitions of $W^c(A_n)$. We determine the unique bijection between $W^c(A_n)$ and the set of non-crossing diagrams of $n+1$ strings that respects the diagrammatic multiplication by concatenation in the $A_n$-type Temperley-Lieb algebra, along with the two algorithms implementing this bijection and its inverse.

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Sadek Al Harbat. 2024-01-17. Catalan numbers: from FC elements to classical diagram algebras. https://arxiv.org/abs/2308.10100

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